Abstract
A p-group is called a C(pw)-group if the normal closure of every non-normal cyclic subgroup has index at most pw. In this paper, we prove that the order of a non-Dedekind C(pw)-group cannot exceed p4w+4 when p > 2 and G′ ≤ HG for every non-normal cyclic subgroup H. We also completely classify non-Dedekind C(p2)-groups for p > 2.
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